Optical Atomic Clocks and the Redefinition of the SI Second: A Review — Epoche C1
The SI second is defined by fixing the frequency of one microwave transition in the caesium-133 atom — the transition between the two hyperfine levels of its ground state — at exactly $9\,192\,631\,770\,\mathrm{Hz}$, a definition adopted in 1967 and retained in the current SI Brochure (BIPM, 2019). The best caesium fountain clocks realise that definition to a fractional uncertainty near $2\times10^{-16}$, which is a drift of one second in $1/(2\times10^{-16}) = 5\times10^{15}\,\mathrm{s}$, roughly 160 million years, and it is tempting to conclude that any further improvement is of academic interest only. It is not. Optical clocks now reach $10^{-18}$, and the consequence is a structural problem: the definition of the unit is worse than the instruments that must report in it, and the barrier to fixing that has turned out to be gravity. Why the unit has become the bottleneck The difficulty is one of traceability rather than of technique. Any frequency quoted in hertz is, by construction, a comparison against the caesium transition, since the hertz is the reciprocal second and the second is that transition. So a laboratory that builds a clock reproducible at $10^{-18}$ and wishes to state its transition frequency in SI units must compare it against a caesium fountain, and the answer inherits the fountain's $2\times10^{-16}$. Nothing about the better clock is lost — the loss occurs at the moment of expressing it. The distinction that matters here, and which the compressed version left implicit, is between an absolute frequency and a ratio. Two optical clocks may be compared directly with each other, and the resulting dimensionless ratio of their frequencies carries no reference to caesium and no caesium uncertainty; ratios between optical standards have been measured well below $10^{-17}$. It is only the absolute value in hertz that is capped. This is why the international programme towards redefinition proceeds through a list of recommended frequencies for secondary representations of the second — optical transitions whose values in hertz are agreed and periodically revised — while the arguments about which optical transition should eventually define the second are settled on ratios (Riehle, 2015). Why a higher frequency wins An atomic clock is a servo loop. A local oscillator — a quartz crystal in a microwave clock, an ultrastable laser in an optical one — is compared repeatedly against a narrow atomic resonance and steered towards it. Its residual instability, measured by the Allan deviation $\sigma_y(\tau)$, the conventional root-mean-square fractional frequency fluctuation remaining after averaging for a time $\tau$, has an ideal limit $$\sigma_y(\tau)\approx\frac{\Delta\nu}{\nu_0}\sqrt{\frac{T_c}{N\,\tau}},$$ in which $\Delta\nu$ is the observed linewidth of the resonance, $\nu_0$ the transition frequency, $N$ the number of atoms interrogated per cycle and $T_c$ the cycle duration. Each factor has a reason, and assembling them is the clearest way to see why optical frequencies win. The measurement in one cycle is of the fraction of atoms driven to the excited state, and the servo operates where that fraction is near one half so that the response to a frequency error is steepest. Each atom is then projected into one of two outcomes with probability near $1/2$, independently of the others, so the number excited is binomial and the fractional uncertainty in the measured excitation is $\sqrt{Np(1-p)}/N = 1/(2\sqrt{N})$ — quantum projection noise, the irreducible statistical floor of the readout. Converting that into a frequency error requires dividing by the slope of the resonance, which is of order $1/\Delta\nu$: a narrower line converts the same readout noise into a smaller frequency error. One cycle therefore yields a frequency uncertainty of order $\Delta\nu/\sqrt{N}$. Repeating for a total time $\tau$ gives $\tau/T_c$ independent cycles, whose average improves as the square root of their number, contributing the factor $\sqrt{T_c/\tau}$. Dividing throughout by $\nu_0$ makes the result fractional, which is the form in which clocks are compared. The decisive term is $\nu_0$ in the denominator, because it is the one quantity that can be changed by five orders of magnitude by choosing a different atom. The strontium-87 clock transition lies at $\nu_0 = 4.29\times10^{14}\,\mathrm{Hz}$, and the ratio to caesium's $9.19\times10^{9}\,\mathrm{Hz}$ is $4.7\times10^{4}$; the aluminium-27 ion transition at $1.12\times10^{15}\,\mathrm{Hz}$ gives $1.2\times10^{5}$, slightly more than five orders. Put numbers through the formula with $\Delta\nu = 1\,\mathrm{Hz}$, $N = 1000$ and $T_c = 1\,\mathrm{s}$. Strontium gives $$\sigma_y(1\,\mathrm{s}) \approx \frac{1}{4.29\times10^{14}}\sqrt{\frac{1}{1000}} \approx 7.4\times10^{-17},$$ and since averaging improves this as $\tau^{-1/2}$, reaching $1\times10^{-18}$ needs an improvement factor of $74$, hence $\tau \approx 74^2 \approx 5{,}500\,\mathrm{s}$: about an hour and a half. The same inputs at the caesium frequency give $3.4\times10^{-12}$, requiring a factor of $3.4\times10^{6}$ and therefore $\tau \approx 1.2\times10^{13}\,\mathrm{s}$, about 370,000 years. The comparison is not about which clock is better made. It is about a factor in a formula. Two constraints govern the other terms and explain why optical clocks were not built in 1970. The observed linewidth cannot be narrower than the Fourier limit set by the interrogation time, roughly $\Delta\nu \approx 1/T$, so a one-hertz line demands that the probing laser stay phase-coherent for about a second — a requirement met only by lasers locked to cryogenic reference cavities. The transition itself must also be narrow enough not to impose a wider limit: the strontium clock transition connects two states of zero electronic angular momentum and is forbidden to first order, becoming weakly allowed in the fermionic isotope $^{87}$Sr through its nuclear spin, which leaves a natural linewidth of order a millih